The experimental crystal structures [1]}, [2]} are used in all calculations.
The nuclei and core electrons are described by the projector augmented-wave potential [3]}, and the wave functions of valence electrons are expanded in a plane-wave basis set with a cutoff energy of up to 520eV.
The spin-orbit coupling (SOC) is included using the second-variation procedure [4]}, [5]}.
In addition, the Hubbard Coulomb interaction \(U={6}{}\)  is included for better accounting for the strong correlation of Eu-\(4f\)  electrons [6]}.
The \(k\) -point integration was performed using a modified tetrahedron method with Blöchl corrections.
A \(16\times 16\times 4\)  \(k\) -point mesh is used for MCAE calculations to ensure sufficient convergence.
